Consider the problem below:

A projectile is launched at 35 degrees above the horizontal and travels 100 meters. Neglecting any outside forces other than gravity, what was the intial velocity v_i along the launch vector?

There are several approaches to solving this problem. Here is one. Follow closely the derivation of the formula for V_i

First, imagine the projectile has JUST landed at point B after travelling for an unknown time. Upon landing, the vertical displacement y at that point in time is ZERO. The distance formula for vertical displacement is
1. and since y = 0,

2.

Now, factor out t from the right side of the equation such that
3. (same formula, right?)
Now think about formula #3. It says that the product of t and the factor in the parenthesis must equal zero. This means that either t is zero, or is zero, or both. After the projectile has arrived at point B, t cannot be zero. This means that IS zero! So,
4. and
5.

Now.. from the basic horizontal velocity formula, we can state that t in the horizontal dimension (which is the same value for t in the vertical dimension in these problems, is

6. . We don't know time in this problem, but it sure comes in handy if we can figure it out. We are going after time next:

We know x, from the problem statement it's 100 meters, but what is the horizontal velocity component v_x? From the diagram about all we can say is that . So lets substitute this value into our time formula #6 above. We see that

7. and this gives us a little better value to use for time. You'll see!

Next, re-examine equation #5. What do we know about v_y ? We know that it also is a component of v_i, that which we seek.
8.
Now check this out: I'm going to do some fine substituting here... I'm going to work with equation 5, and substtute equation #7 for time (t), and substitute equation #8 for v_y
9.

Multiply both sides of equation 9 by and we now have (don't go away.. almost there!)
10. or
11. . Now divide both sides by and we have the Grand Finale
12.

Write this dude down! This is a very usable formula for solving for vi (don't forget to take the square root in formula 12!) when only angle theta and the range (x) are known.

Use this relationship to solve problems 43 and 48 at the end of chapter 7!

Here is a summary of this page in pdf format.

bunning '98